Feb., 1940
RELATION BETWEEN SOLUBILITY AND DIELECTRIC CONSTANT
407
[CONTRIBUTION FROM THE DEPARTMENT O F CHEMISTRY, N E W YORK UNIVERSITY]
An Empirical Relation between Solubility of Slightly Soluble Electrolytes and Dielectric Constant of the Solvent BY J. E. RICCIAND T. W. DAVIS many solvents. The quantity - log f*, for thallous iodate, calculated from its solubility in alcohol-water solvents, on the basis of the limiting law of Debye and Hiickel, by La Mer and Goldman; is practically a constant. The mean activity coefficient, f*, of a number of uni-univalent electrolytes in pure methanol was determined If the interionic effects are taken into account, by Bronsted, Delbanco and Volqvartz,6 on the by introducing the Debye-Hiickel expression for basis of solvent effects of added salts, and was the mean ionic activity coefficients of the electro- noted to be practically the same as that in satulyte in each medium, the solubility being suffi- rated aqueous solutions; although this fact was ciently low in each, the equation is modified as specifically mentioned by these authors, no further notice was taken of it, or of any possibility of follows applying the relation. From the measurements of Bronsted and Williams’ on the solubility of complex cobalt ammines in the presence of added salts, the activity coefficient of croceo tetranitro-diammino cobaltiate (a uni-univalent electrolyte) is calwhere A , B and C are constants a t any given culated to be 0.981 a t saturation both in pure temperature, S the solubility, D the dielectric con- water (D = 81.1) and in water saturated with stant, the 2’s represent ionic valences, both con- ether (D = 74.5)) a t 18’; and that of the luteo sidered as positive numbers, ,LL the ionic strength, salt (a 3:l type), 0.916 and 0.907, respectively, in r the ionic radius in the Born sense, and a the same solvents. Similar measurements by the distance of closest approach of the ions in the Hansen and Williams8 give the following values for Debye-Hiickel sense. The subscripts refer to the f* a t saturation, for the croceo cobaltiate a t 25’ two different media.3 Equations 1 and 2, both, (a) in water (D = 7 8 . 5 ) ,0.978 require previous knowledge of the ion size a,and (b) in 20 mole $7, ethanol (D = 54.0), 0.969 in the absence of such information the equations (c) in 40 mole % ethanol (D = 41.4), 0.969 cannot be used. (d) in 60 mole % ethanol (D = 33.8), 0.974 I n connection with studies of the solubilities of The values for the croceo sulfate (a 1 : 2 salt) salts in dioxane-water mixtures,2 we have noted are 0.805, 0.929 and 0.952 in (a), (b) and (c), rethat the activity coefficient of an electrolyte in spectively, and for the luteo iodate (3:1), 0.537, saturated solution is in many cases practically con0.756 and 0.857, respectively, in the same solstant and independent of the nature of the solvent. vents. We see here a substantial constancy a t Several examples of this empirical regularity are least in the case of the simpler types. found in the literature although no especial sigTable I shows the mean activity coeEcients for nificance has hitherto been laid upon it. Thus silver acetate in its pure saturated solutions in Walden4 reported tetraethylammonium iodide to various solvents, as determined by the usual be about 48% dissociated in saturated solution in method of solvent salt effects. The approximate (1) M. Born, Z.Physik, 1, 45 (1920). constancy off is evident. (2) T. W. Davis and J. E. Ricci with C. G.Sauter, THISJOURNAL,
The solubilities of an electrolyte in two media of different dielectric constant are related, on the basis of the Born theory of ion-solvent interaction, and ignoring ion-ion interactions, by the equation2
61, 3274 (1939). (3) While “a” and “2r” are both ionic diameters, they apply to different processes and might have quite distinct values for a given salt. We have regarded this as a minor question, however, in this paper, and hereafter equate “a” and “2r” at least as a first apa proximation. (4) P. Walden, 2,physik. Chcm., 66, 683 (1906)t
(5) V. R. La Mer and P. H. Goldman, THIS JOURNAL, 63, 473 (1931). (6) J. N. Bronsted, A. Delbanco and K. Volqvartz, Z. ghrsik. Chem., 1688, 128 (1932). (7) J. N. Brbnsted and J. W. Williams, THISJOURNAL,60, 1338 (1928). ( 8 ) L. A. Hansen and J, W, Williams, ibid., 6P, 2769 (1980).
J. E. RICCIAND T. W. DAVIS
408
Vol. G2
TABLE I medium. In equation 4 the variation of the ACTIVITY COEFFICIENT OF SILVERACETATE IN PURE solubility with D is seen to be independent of the SATURATED SOLUTION, AT 25' valence type of the electrolyte since the valence Dielectric factors cancel out of the equation. In equation Solvent constant ft Watery 78.55 0.800 5, the valence factors again cancel except those inAlcohol9 l ( J % 72.9 .797 volved in the definition of the ionic strength in the 20 % 67.1 ,803 denominators, so that the effect of the valence 30 % 61.2 ,802 type is only very slight. Acetonelo 10% 73.0 .SO4 Neglecting effects of ionic size, and using, there20% 67.0 ,805 30% 61.0 ,798 fore, equation 4, the solubility in the second meDioxaneZ 10% 69.7 .786 dium is related to that in the first or reference 20% 60.8 ,771
If this observation should be found to apply generally even though approximately, i t offers a simple way of predicting solubilities in any medium from a knowledge of the value in one, inasmuch as the logarithm of the activity coefficient a t saturation is related directly to the ionic strength (and hence to the saturation concentration for a pure solution if complete dissociation is assumed), by means of the familiar Debye-Hiickel equations. If the activity coefficient of an electrolyte, a t saturation, is a constant, independent of the dielectric constant of the medium, the solubilities in two media will be related, where the DebyeHiickel limiting law holds, by the equations:
- logf, loe ( Z + Z - ) ~ / W ' / Z 013/2 l/z
1.8123 X
=
- logf2 -
1.8123 X lo6 ( z + z - ) ' / d / s D ~ ~T/ ~z /l/z Z
(3)
l/s;
(4)
where T is the temperature and v is the total number of ions formed by the complete ionization of one molecule of the electrolyte, and the other symbols have the meanings previously noted. If the Debye-Hiickel expression for the activity coefficient is modified to take into account the effect of the ionic size, then the equation takes the form
where B = 1.8123 X lo6 (z+z-)'/' v ' / ' / f i , and A = 50.288 (Z+Z-)'/' Y'/' 1/2, a being in Angstrom units. In either case the solubility in any dielectric can be calculated from that in any one reference (9) F. H.MacDougall and C. E. Bartsch, J. Phrs. Chem., 40, 64B
(leas).
(19) F. H.MacDougalland W . D. Larson, ibid., 41,417 (1937).
medium at the same temperature, by the equation Sl'/z / Di '//n = S ~ ' / D2'/2 P (6) or log St
log SI
+ 3 (log Dz - log D I )
(7)
so that a plot of the logarithm of the solubility against the logarithm of D would give a straight line with a slope of 3. The solubility according to equation 5 can be written in the form log Ss = log
("> + log Dz + log T - 2 log [1.8123 - 50.288 K a ] uz+z-
Z+Z-
(8)
where K is the negative logarithm of the activity coefficient in the first medium. In order to determine the extent to which this empirical rule of constant activity coefficient may be applicable, and to test its possible usefulness in correlating known solubilities and hence in predicting unknown solubilities, we have examined much of the published data on solubilities of slightly soluble electrolytes in non-aqueous solvents and in mixtures of water and organic liquids. For comparison between the calculated and the observed solubilities (Sobsd,), the calculations were made, from the solubility in water, according to the various principles : thus, Sf represents the solubility calculated by means of the simplest equation, namely, eq. 7; Sfa,that according to equation 8; and SB,,,, according to equation 1. The latter two calculations of course could be made only where suitable values of the ion diameters could be found. The values of a used in the various cases were taken from reports in the literature on aqueous solutions, some determined by solubilities, some by e. m. f.'s. The dielectric constants for the various solvents were taken from the literature, principally from the data of Akerlof." (11) G. Akerll)f, THISJOURNAL, 54, 4126 (1932); G .Kkerlof and 0 A. Short, ibid., 18, 1241 (1936).
RELATION BETWEEN SOLUBILITY A N D DIELECTRIC CONSTANT
Feb., 1940
As an example and to facilitate comparison, the data and calculations for the solubility of barium iodate monohydrate in dioxane-water solvents,2presented in Table 11, are shown graphically also in Fig. 1, where log S has been plotted against log D. The predictions of the constant activity coefficient rule, as can be seen both from the table and from the curves, are a t least in moderate agreement with the observed values over the whole range, while the Born equation very quickly predicts much too small solubility. Since Sja differs so little from S, in this case, values of Sfohave not been calculated for the other salts reported, even where values of a were available.
1.9
1.7
log 1.5
409
D. 1.3
1.1
0.9
0.7
TABLE I1 SULLWLITY OF BARIUM IODATE MONOHYDRATE IN DIOXAXE-TvATER SOLVENTS, AT 25'; a = 2.34 A.(')
8.145 4.687 2.407 1.167 0.5198 . 2550 ,1247 . ( IY:%:?
,0170 ,0020 .O'~l10
5.66 3.76 2.34 1.33 0,673 ,289 ,0926 .020(i
00203 (IO( !1 (i
5.60 3.68 2.27 1.28 0 643 ,274 0872 ,0192 00275 00014
1.2 1.6 2 ,0 2.6 2.6 2.3 1.0 1.2 1. 0
'l'ht tlegree of usefulness of the principle of constant activity coefficient a t saturation, in predicting a t least the order of magnitude for solubilities i n orgatiic sol\,ent5 from the Anown solubility i n water, is best judged by nieans of the ratio, .Sf' .50,,\,l , especially if this is compared with t h e w n t : ratio for thc solubility calculated from the Horn equatioii. Thus in the case of barium iodate moiwhydrate, the ratio, SflSobsd , is correct throughout as far as the order of magnitude j, concerned, within a maximum factor of only 2.6, whiie the ratio SRorn/Sob~d , becomes extremely low especially in the high dioxane mixtures. The results of the comparison of observed and calculated solubilities for a number of uni-univalent electrolytes in various solvents are shown in Tables 111, IV and V. Table I11 lists the results for silver acetate in dioxane-water solvents; here the ratio, Sf/SObsd , is, throughout the very wide range of D covered, again satisfactory in respect to the order of magnitude of the calculated solubility, the factor being l/b at worst. Table
Fig. 1,-Solubility of Ba(I08)2.H20 in dioxane-water mixtures: the straight line, as predicted on constant activity coefficient; the curve, as predicted by the Born equation; the points, as observed, 0 .
IV gives the ratio, Sf/Sobsd., for a number of I : 1 electrolytes in mixtures of water with various organic substances. The value of the ratio is seen to be such, throughout, as to predict at least roughly the solubility in the various media. The dielectric constants represented in this group vary from 112.4 in the case of silver chloride in glycine solutions, to that of ethanol a t 40°, 2 2 . 2 ; a variation of temperature, between 14 and -lOo, is also includerl, i n the data for the perchlorates in ethanol-water inixtiircs. l'able V givtbs TABLU 111 SOLURILITY
(.IF SIL\'liR
ACETATE
SoLvEwrs A i 250;n \vt % clioxnne
0 10 20
30 40 50 60 70 80 90 100
=
1.U I ~ I ( . J X A K I + - ~T~RAW
:i.sn A.(z)
.'Ob.
